A097165 -id:A097165 - cp's OEIS frontend

A097164 Expansion of (1+3x)/((1-x)(1-4x^2)).

1, 4, 8, 20, 36, 84, 148, 340, 596, 1364, 2388, 5460, 9556, 21844, 38228, 87380, 152916, 349524, 611668, 1398100, 2446676, 5592404, 9786708, 22369620, 39146836, 89478484, 156587348, 357913940, 626349396, 1431655764, 2505397588

Offset: 0

Paul Barry, Jul 30 2004

Partial sums of A084221. a(n) = A097163(n+1)/4. Third binomial transform is A097165.
a(n+1) = 4*A097163(n). - Zerinvary Lajos, Mar 17 2008
See A133628 for an essentially identical sequence. - R. J. Mathar, Jun 08 2008

Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=4*a[n-2]+4 od: seq(a[n], n=1..31); # Zerinvary Lajos, Mar 17 2008

CoefficientList[Series[(1+3x)/((1-x)(1-4x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,4,-4},{1,4,8},50] (* Harvey P. Dale, Jul 11 2023 *)

a(n) = 5*2^n/2 - (-2)^n/6 - 4/3;
a(n) = a(n-1) + 4a(n-2) - 4a(n-3).
G.f. ( 1+3*x ) / ( (x-1)*(2*x+1)*(2*x-1) ). - R. J. Mathar, Jul 06 2011

A097162 a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2))*2^k.

Original entry on oeis.org

1, 3, 7, 21, 37, 123, 187, 681, 937, 3663, 4687, 19341, 23437, 100803, 117187, 520401, 585937, 2667543, 2929687, 13599861, 14648437, 69047883, 73242187, 349433721, 366210937, 1763945823, 1831054687, 8886837981, 9155273437, 44702625363

Offset: 0

Paul Barry, Jul 30 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-20,20)

Cf. A075427.

Cf. A057651, A097165.

A097162:=n->add(binomial(floor((n+1)/2),floor((k+1)/2))*2^k, k=0..n): seq(A097162(n), n=0..30); # Wesley Ivan Hurt, Sep 18 2014

LinearRecurrence[{1,9,-9,-20,20},{1,3,7,21,37},50] (* Vincenzo Librandi, Jan 30 2012 *)

G.f.: (1+2*x-5*x^2-4*x^3)/((1-x)*(1-4*x^2)*(1-5*x^2)).
a(n) = (3/4-3*sqrt(5)/4)*(-sqrt(5))^n +(3/4+3*sqrt(5)/4)*(sqrt(5))^n-(2^n-(-2)^n)-1/2.
a(2*n) = A057651(n); a(2*n+1)=3*A097165(n).
a(n+5) = 20*a(n)-20*a(n+1)-9*a(n+2)+9*a(n+3)+a(n+4). - Robert Israel, Sep 18 2014

A155485 a(n) = 5^n + (1 - 4^n)/3.

Original entry on oeis.org

1, 4, 20, 104, 540, 2784, 14260, 72664, 368780, 1865744, 9416100, 47430024, 238548220, 1198333504, 6014037140, 30159664184, 151156234860, 757212830064, 3791790773380, 18981860359144, 95000927764700, 475371142699424, 2378321729000820, 11897472707018904

Offset: 0

Paul Curtz, Jan 23 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -29, 20).

[5^n+(1-4^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 07 2011

a(n) = A000351(n) - A001045(2n).
a(n) = 10a(n-1) - 29a(n-2) + 20a(n-3), as in A097165.
From R. J. Mathar, Feb 27 2009: (Start)
G.f.: (1-3x)^2/((1-x)(1-4x)(1-5x)).
a(n) = A097165(n) - 3*A097165(n-1). (End)

Edited, Lava definition adopted, and extended by R. J. Mathar, Feb 27 2009

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A097164 Expansion of (1+3x)/((1-x)(1-4x^2)).

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A097162 a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2))*2^k.

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A155485 a(n) = 5^n + (1 - 4^n)/3.

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